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Integral transforms based upon fractional integration

Published online by Cambridge University Press:  24 October 2008

Charles Fox
Charles Fox
Affiliation:
McGill University, Montreal, Canada
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Abstract

The theory of Fourier transforms

can be developed from the functional equation K(s) K(1 – s) = 1, where K(s) is the Mellin transform of the kernel k(x).

In this paper I show that reciprocities can be obtained which are analogous to the Fourier transforms above but which develop from the much more general functional equation

The reciprocities are obtained by using fractional integration. In addition to the reciprocities we have analogues of the Parseval theorem and of the discontinuous integrals usually associated with Fourier transforms.

In order to simplify the analysis I confine myself to the case n = 1 and to L2 space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 1 , January 1963 , pp. 63 - 71
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Bochner, S.Inversion formulae and unitary transforms. Ann. of Math. 55 (1934), 111–115.CrossRefGoogle Scholar
(2)Erdélyi, A.On some functional transforms. Rend. Sem. Mat. Univ. Politech. Torino, 10 (1950–1951), 217–234.Google Scholar
(3)Fox, C.A classification of kernels which possess integral transforms. Proc. Amer. Math. Soc. 7 (1956), 401–412.Google Scholar
(4)Hardy, G. H. and Titchmarsh, E. C.A class of Fourier kernels. Proc. London Math. Soc. (2), 35 (1933), 116–155.CrossRefGoogle Scholar
(5)Kaczmarz, S.Note on a general transform. Studia Math. 4 (1933), 146–151.CrossRefGoogle Scholar
(6)Kober, H.On fractional integrals and derivatives. Quart. J. Math. Oxford Ser. 11 (1940), 193–211.Google Scholar
(7)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford, 1937).Google Scholar
(8)Watson, G. N.General transforms. Proc. London Math. Soc. (2), 35 (1933), 156–199.Google Scholar
(9)Whittaker, E. T. and Watson, G. N.A course of modern analysis (Cambridge, 1915).Google Scholar